Notes for Measure Theory

I am not a mathematician, my only higher level mathematics course was, what would now be called, a service course. I took this course over 40 years ago and the approach was very practical. So as I work through the book on measure theory I am having to look up definitions which would probably be obvious to a newer mathematics graduate.

This page records those definitions.

The Axiom of Choice – if a set {A} is partitioned into a number of non-empty disjoint subsets, {P_i}, the axiom of choice says that we can always obtain a set, {T}, by choosing one element from each of the {P_i}. This seems self evident when the number of subsets is finite; the axiom of choice extends this to the case where the number of subsets is infinite.
Countable – a set, {A}, is countable if one can define a bijective mapping from {A} to a subset of the set of natural numbers, {\mathbb{N}}. If the mapping is from {A} to {\mathbb{N}} then {A} is said to be countably infinite.
Infimum – given a set {A}, the infimum, {\inf A}, is the greatest lower bound of {A}. For example, if {A=\{1/n: n\in \mathbb{N}\}}, there is no minimum value, but {\inf A} is defined and equals {0}. Note that {\inf A \notin A},
Supremum – given a set {A}, the supremum, {\sup A}, is the least upper bound of {A}. For example, if {A=\{1-1/n: n\in \mathbb{N}\}}, there is no maximum value, however {\sup A} is defined and equals {1}. Note that {\sup A \notin A},
Uncountable – a set {A} is uncountable if, and only if, it is not finite and not countable. The set of real numbers {\mathbb{R}} is uncountable.

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